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Also, these two methods are quite convenient to analyze 2D problem of failures. 1. Several modeling examples show that shear strain, displacement, tensile failure location, can be calculated at the same place in the slope as observed in the field (Agliardi et al., 2001, 2009a; Ambrosi and Crosta, 2011; Baro et al., 2005; Brideau et al., 2006; Eberhardt, 2008; Eberhardt et al., 2004; Kinakin and Stead, 2005; Sausgruber et al., 2010). Open folds may show the typical transverse, longitudinal, and conjugate discontinuities associated with folding, whereas tight and isoclinal folds may develop foliation and cleavage, a sign of increased damage. Discontinuity sets associated with folding (i.e. S0=bedding, S1=discontinuity set 1, S2=discontinuity set 2. a) Bedding dips parallel to slope and orthogonal discontinuity set acts as rear release, b) bedding dips into slope and orthogonal discontinuity set acts as main sliding surface, and c) slope face intersects with fold hinge and bedding and orthogonal discontinuity set create a biplanar failure surface (modified from Badger, 2002). Doug Stead, Andrea Wolter, in Journal of Structural Geology, 2015. Plane failure of rock mass occurs when the discontinuities are parallel to rock slope and dips toward the slope face. Figure 6.7. The characterisation includes the mechanical properties of intact rock, orientation and surface characteristics of discontinuities, geologic history of the rock mass, climate and topography of the slope and any past and currently active tectonic and geomorphic processes. As Bovis (1982) pointed out, although steepening and debuttressing occurred in every glaciated valley, not every slope has failed, and he concluded, as others had done (e.g. A number of studies have demonstrated the rock avalanche/mass flow origin of coarse rubbly deposits in valley fills (Porter and Orombelli, 1980; Fort, 1987, 2000; Fort and Peulvast, 1995; Hewitt, 1999; Fauqu etal., 2009) in a reinterpretation of deposits previously mapped as morainal material. The notion of representative volume element (RVE) of jointed rock masses was proposed by Pariseau et al. For each situation, three different dip angles (30, 50, and 70) were used. In both approaches, the rock mass is discretized into a finite number of elements. ScienceDirect is a registered trademark of Elsevier B.V. ScienceDirect is a registered trademark of Elsevier B.V. They assumed the horizontal top surface of rock slope and analysis was based on Mohr-Coulomb shear strength criteria. More recently, the 2012 Gayari icedebris avalanche, Pakistan (Schneider etal., 2013) buried a Pakistan army camp with the loss of 128 soldiers (Figure16.5). Recently, fast progresses in DEM techniques allow performing detailed study of complex rock slope situations requiring the application of hydro-mechanical, thermo-mechanical and dynamic approaches. However, as Huggel etal. 13c), respectively. CT=Col Tramontin Fault, and CE=Col delle Erghene Fault in a). I have run Mathcad for the fracture pattern at the lower right of Figure 6.9 for several p directions and graphed it in a polar plot to make it easier to visualize the fracture tensor (left polar plot of Figure 6.11). Tectonic effects on roughness, such as slickensides and polishing, have been recognised as indicators of residual strength and the need for testing the shear strength in the appropriate direction with respect to surface structures noted. The influence of tectonic activity may lead to a reduction in the interbed shear strength with an important influence on both sliding and toppling movements. Roughness exerts a primary control on rock slope failures and exists at all scales from primary and secondary small-scale asperities to large-scale undulations. In the 1961 Peru Census, the population of Yungay was recorded as being 3,543, a population density of 9,980 p/km2. An understanding of the multiple attributes, contributing factors and processes is required to conduct a comprehensive hazard characterisation of bedrock landslides. Rock slope stability analysis has always been a topic of immense interest for researchers and geotechnical practitioners because of its safety and economical damages. 2013 ASTER image of the city of Huaraz in the Peruvian Andes. In addition, probabilistic slope stability analysis and probabilistic back-analysis was applied to landslides which occurred at the Arakli-Tasonu quarry, NE Turkey, on October 3, 2005, March 20, 2006, and October 19, 2006. 5. Note steepening of bedding adjacent to fault in b), which resulted in additional driving forces for failure and ultimately slope instability. You can easily program it in C, C++, Fortran, Matlab, or some other software program. Folding in two directions and the presence of interference folds has been shown to influence the complexity of slope deformations within landslides and, in combination with natural release surfaces, may explain commonly observed multiple block-type movement. Under the purview of stability analysis of a rock slope, the strength values are, in recent years, been regarded as a variable in the range from the peak strength to the residual strength (Bhattacharya, Chowdhury, & Metya, 2019; Metya, Bhattacharya, & Chowdhury, 2016a, 2016b), as because the sliding failure of a rock mass is generally a progressive failure process (Metya, Bhattacharya, & Chowdhury, 2016c; Metya, Dey, Bhattacharya, & Chowdhury, 2019; Pan, Sun, Wu, & L, 2017; Scholts & Donz, 2012) and the softening behavior of a rock mass is also been demonstrated through a series of laboratory tests (Casagrande, Buzzi, Giacomini, Lambert, & Fenton, 2018; Sow et al., 2017; Tiwari & Latha, 2019). These issues await further exploration (cf. An illustrative example was presented by Hencher (2006), in which progressive development of sheeting joints over a period of many years was observed prior to the detachment of a large landslide in Hong Kong. Figure 6.8 shows such an imaginary sphere. The range of bedrock mass movement types within the Varnes-based landslide classification scheme is also reviewed. Aerial photography can provide help in this endeavor by delineating areas of past rock mass failures as with soil slopes, but we can provide more help than that by providing quantitative, if only limited, information on the character of the blocks and discontinuities that form a rock mass. Fig. The landslide movements in soil-like material have been modeled showing the ground surface displacements, but most of the time the vertical deformation is integrated (Vulliet and Hutter, 1988), which does not provide the detailed deformation perpendicular to the flow. 9. a) Example of drag folding-induced slope failure at a former UK coal mine. This 3D FDM model, considering cohesion weakening and frictional strengthening, demonstrates the creation of shear failure at the toe and tensile failure high on the slope. The main contribution of RVE approach is that numerous non-persistent discontinuities within a rock mass at project scale can be effectively dealt with. (2008) aiming to simultaneously enhance the reliability of large-scale rock mass stability analysis and dramatically reduce computer run time, from hundreds of hours to several hours. In this study, rock slope stability analysis, especially probability analyzes will be reviewed. Hoek and Bray (Hoek & Bray, 1981) gave an analytical expression for the factor of safety of rock mass against the planar mode of failure using a limit equilibrium approach. Nevertheless, computer power is progressively increasing allowing for higher model resolution, permitting a continuum approach to be adopted for most of the problems (Podladchikov, pers. Patton's demonstrated the influence of roughness on shear strength by conducting a direct shear test on a saw-tooth specimen (Fig. The site of Yungay before (a) and after (b) the May 31, 1970, debris flow in georeferenced aerial photographs (Evans etal., 2009a). Both tensile and shear cracks occurred when centre of gravity was in the lower part of block. Focus in rock engineering has been on the use of roughness measures such as the Joint Roughness Coefficient, JRC (Barton, 1976, 2011; Barton and Choubey, 1977). Transverse, longitudinal, and converse discontinuities may be present in open folds, whereas tight folds may develop foliation and cleavage. Among these, Ling and Cheng (Ling & Cheng, 1997) also gave the formulation using coulomb criteria for large width using plane strain conditions. 6b; Massironi etal., 2013), but also by concretions on the centimetre scale. Sharma et al. Discontinuum approaches examine the problem assuming an assemblage of rigid or deformable blocks, exposed to gravity or external loads allowing a more accurate description of failure mechanisms in which block connectivity extensively changes (Jing, 2003). Fischer et al., 2006), as well as the direct effects of atmospheric temperatures. The orientation of the topographic surface or excavation face with respect to bedding is of major significance. All we wish to do is provide some indication of the directions at which rock blocks are larger or smaller. Both analyses were done using the fracture network shown in the lower right of Figure 6.9.

As a general rule, the permeability within a rock mass is controlled by flow through the fractures. Warming may result from changes in snow and ice cover ice during the retreat of glaciers (c.f. Depending upon photo scale, we may not be able to see all of the large fractures, or any of the small fractures, on aerial photographs. This surface is affected not only by two generations of regional folds, producing complex, metre-to decametre-scale dome-and-basin to crescent-and-mushroom 12 (Ramsay, 1967) or K (Thiessen and Means, 1980) type interference patterns (Fig. The critical path for a given discontinuity geometry (including coplanar and non-coplanar joint planes, such as en enchelon) was defined to consider strength contributions from discontinuities and intervening rock bridges as well as the spatial variability of discontinuity geometry. J. Shang, Z. Zhao, in Engineering Geology, 2018. Many rock slope stability problems involve complexities relating to geometry, material anisotropy, nonlinear behavior, in situ stresses, and the presence of several coupled processes (pore pressures, seismic loading, and etcetera) (Chiwaye, 2010). Assuming there are m fracture sets, the fracture tensor in 3-D is defined as, In the above equation, is the fracture density (number per unit volume), r is the radius of a circular fracture, ni and nj are the outward normal components in the i and j directions, f(n,r) is the joint probability density function for n and r, and /2 is the solid angle corresponding to the surface of a unit hemisphere (radius=1). 9b shows the fault exposed within the mine and the drag folding. I have used an upper hemisphere projection here for clarity. Only one set of parallel discontinuities with varying persistence was examined in their study. Non-persistent rock discontinuities have significant influence on the mechanical properties and deformability of rock masses and therefore on the stability of rock engineering projects such as engineered rock slopes. For higher dip angles, rock slopes often fail in by both tensile and shear cracking. A factor of safety model is developed using Patton's theory for the factors and cases as mentioned above. Fold and slope interaction creating slope instability (indicated by grey areas). Another example of structural analysis associated with natural slope failures is that conducted by Humair etal. Rock-slope stability and long-term equilibrium slope angles are controlled primarily by rock-mass properties. According to Lorig and Varona (2004), the main contributions of numerical models in rock slope stability analyses can be summarized in four main points: automatic extrapolation of the potential failure mechanism(s) and the most probable failure surface (or shear zone); implicit or/and explicit incorporation of significant geological features (e.g., faults, weathered zones) and groundwater conditions providing more realistic analysis than classical limit equilibrium approaches; more detailed analysis about the physics of phenomenological behavior of the slope instability; possibility of testing multiple topographical or geological conditions and failure mechanism situations, and proposition of different design options. Furthermore, we can find no information on fracture dips. Michel Jaboyedoff, Giovanni B. Crosta, in Tectonophysics, 2013. Indeed, the broad paradigm that has developed in the last 20 years linking climate change (and associated cryospheric response) to CMFs involves the creation of conditions more favorable for their occurrence and an increase in their frequency (e.g., Evans and Clague, 1993, 1994; O'Connor and Costa, 1993; Haeberli etal., 1997; Abele, 1997; Huggel, 2009; Huggel etal., 2010, 2012; Stoffel and Huggel, 2012; Allen and Huggel, 2013). We can, however, use low order streams (see, for example, Figures 3.10 and 3.11Figure 3.10Figure 3.11) to infer fracture orientations and lengths where we cant actually see the fractures themselves. The 1941 Huaraz disaster resulted in the deaths of approximately 4,000 persons when a mass flow cut through the northern segment of the city (Figures 16.2 and 16.21) destroying about 1km2 of densely populated urban terrain (Oppenheim, 1946; Heim, 1948; Carey, 2005, 2010; Evans and Delaney, unpublished data). Stephen G. Evans, Keith B. Delaney, in Snow and Ice-Related Hazards, Risks, and Disasters, 2015. As mentioned above, modeling directly the development of observed gravitational features such as faults and folds remain a difficult task. Three events in the Cordillera Blanca of Peru resulted in the deaths of about 12,000 people between 1941 and 1970. So the projections of their normal lines will tend to form clusters as shown in Figure 6.9. Limited work in rock slope engineering has been undertaken to date considering fault contact relationships and their influence on rock slope instability; this includes not only damage but also fault linkage, drag mechanisms, fault seals, shale smear, the nature of fault gouge, cataclasis, and diagenetic effects. (2010); Fig. Figure 6.10. In the analysis of initial failure, in addition to the geological and geotechnical characterization of the source rock mass, quantification of the role of the degradation of high-altitude permafrost in, Gruber and Haeberli, 2007; Fischer etal., 2010; Krautblatter etal., 2013, Evans etal., 2009a; Schneider etal., 2010, Reynolds etal., 1998; Carey etal., 2012, Oppenheim, 1946; Heim, 1948; Carey, 2005, 2010, Haeberli etal., 2004; Huggel etal., 2005; Evans etal., 2009b, Hauser, 2002; Evans etal., 2009a; Huggel etal., 2012, Thorarinsson et al., 1959; Rapp, 1960; Beck, 1968; Bovis, 1982; Augustinus, 1992, 1995a; Alexandrowicz, 1997, Slope Tectonics: Structures and Slope Failures, Brideau et al., 2009; Eberhardt et al., 2004; Stead et al., 2006, , the main contributions of numerical models in, Agliardi et al., 2001, 2009a; Ambrosi and Crosta, 2011; Baro et al., 2005; Brideau et al., 2006; Eberhardt, 2008; Eberhardt et al., 2004; Kinakin and Stead, 2005; Sausgruber et al., 2010, Agliardi et al., 2001; Chemenda et al., 2009; Kinakin and Stead, 2005; Leith, 2012, Forlati et al., 2001; Ghirotti et al., 2011; Stacey et al., 2003, Preh et al., 2003; Stead and Coggan, 2012; Stead et al., 2012, Bachmann et al., 2009; Chemenda et al., 2009; Bretschneider et al., (2013--this issue), Sausgruber et al., 2010; Zischinsky, 1969, Permafrost and climate in Europe: Monitoring and modelling thermal, geomorphological and geotechnical responses, Thus, many processes (operating at different temperatures and time scales) may translate warming or thaw of permafrost into reduced, Geological discontinuity persistence: Implications and quantification, proposed a probabilistic criterion for failure that was related to discontinuity data, to examine the effect of discontinuity persistence on, A critical review of rock slope failure mechanisms: The importance of structural geology, , should be considered with respect to how they influence, Barton, 1976, 2011; Barton and Choubey, 1977. There are so many methods available in the literature for assessing the stability of a rock slope, such as the kinematic method, the empirical method, the limit equilibrium method (Goodman, 1988; Hoek & Bray, 1981; Karaman, Ercikdi, & Kesimal, 2013), the upper and low bound method (Chen, 2004; Zhou & Wang, 2017), the finite element method (Baba, Bahi, Ouadif, & Akhssas, 2012; Eberhardt, 2003; Jiang, Qi, Wei, & Zhou, 2015; Wang, Yin, Chen, & Lee, 2004) and the discrete element method (Son & Adedokun, 2015; Stead, Eberhardt, & Coggan, 2006).

Following are the conditions of plane failure: (a) discontinuity plane must strike slope face within 20 degree and (b) angle of failure plane must be more than angle of internal friction (Fig. A second implication of the complexity relates to the modeling of both initial failure and runout for hazard assessment (Huggel etal., 2004a,b). Rock slopes fail generally into these five modes namely planar, wedge, circular, toppling, and buckling (Hocking, 1976; Hoek & Bray, 1981; Kliche, 1999; Lee & Wang, 2011; Ramamurthy, 2014; Tang, Yong, & Ez Eldin, 2017). FDM technique was also applied to estimate stress conditions and displacement patterns for DSGSD both at a local (Agliardi et al., 2001; Chemenda et al., 2009; Kinakin and Stead, 2005; Leith, 2012) and to infer deformation mechanism at regional scale (Martino et al., 2004). Applications of the progressive failure concept are presented to explain the behaviour of rock masses leading to a catastrophic slope failure. Shear testing of Saw-tooth specimen (A) Shearing on a saw-tooth specimen and (B) Patton's shear strength graph. A key to analyzing rock mass failure is achieving an understanding of the geometry of rock mass discontinuities. Whether the folding is of a parallel or concentric nature or similar/shear type will also influence kinematics and potential failure block geometry. I have cleverly restricted consideration to 2-D. The Fry diagram3 is a center-to-center spatial analysis designed by Fry to represent nonisotropy in mineral crystals as an indicator of strain. (Sharma et al., 1995) modified the approach for inclined top and inclined tension crack. 7. Figure 6.9. Limit equilibrium method and strength reduction method are generally used to determine the factor of safety. Later on, many researchers proposed the limit equilibrium approach and modified the Hoek's model for various practical possible situations (Ahmadi & Eslami, 2011; Hoek, 2007; Ling & Cheng, 1997; Price, 2009; Sharma, Raghuvanshi, & Sahai, 1999; Shukla, Khandelwal, Verma, & Sivakugan, 2009; Tang et al., 2017; Zheng, Liu, & Li, 2005). The debris flood/debris flow originated in the outburst of Lake Palcacocha 22km upstream, which developed into a destructive high-velocity flow (direction of flow indicated by arrows). Mohr-Coulomb assumed the smooth failure surface similar to soil. (2011) based on detailed remote sensing analysis and 3D geodetic patterns performed a 2D numerical analysis to reproduce the hypothesized kinematic behavior of the Randa rockslide (Switzerland). The Fry approach is best applied using 300 or more rock blocks. Charles E. Glass Ph.D., P.E., in Interpreting Aerial Photographs to Identify Natural Hazards, 2013. The fracture tensor is dominated by the longer fractures in the northwestsoutheast direction, as you would expect. (a) the urban area of Yungay is outlined by a white line. Stead et al. The RVE of a non-persistently jointed rock mass represents the smallest volume over which a measurement can be made that will yield a value representative of the whole. In the Huascarn mass flows of 1962 and 1970, Peru, it was established (Evans etal., 2009a) that previously reported death tolls (25,000 in the case of the 1970 event) were overestimated by a misinterpretation of the 1961 Peru Census, a possibility first raised by Clapperton and Hamilton (1971). The Huascarn events illustrate the need for a careful and objective analysis of the human impact of destructive CMFs in the glacial environment and highlights the risk to which a local population is exposed when CMF debris is reoccupied for housing and agriculture despite the fact that conditions appear to exist for a recurrence of a CMF originating once more on the North Peak of Huascarn (Evans etal., 2009a).

In the second case, aerial photographs can be combined with other techniques to help simulate an equivalent porous media (EPM). 4ivvii) may result in complex translational and rotational slope displacements in both planar and toppling modes (Brideau, 2005; Yan, 2008). (1983) proposed a probabilistic criterion for failure that was related to discontinuity data, to examine the effect of discontinuity persistence on rock slope stability. Bold arrows show the direction of block movement down the failure plane or wedge intersection. In this chapter, Hoek's model is modified for the inclined free surface of rock slope and considered other factors such as partially water filled up tension crack, rock bolt stabilizing force, horizontal and vertical seismic forces. The most common approach to describing the fracture patterns within a rock mass is to project normal lines from individual fractures onto the lower hemisphere then back up to a 2-D projection of the equatorial plane. As Allen etal. The dashed lines correspond to tensile crack while continuous lines represent shear crack. Just remember that engineering practice usually uses a lower hemisphere projection. Thorarinsson et al., 1959; Beck, 1968), that lithology and rock structure are (pre-conditioning) factors that provide ultimate control of stability and the distribution of rock slope failures. Continuum numerical techniques commonly used in slope analyses include finite-element (FEM) and finite difference (FDM) approaches. Based on instrumental observations and numerical modeling results, they suggested that seasonal thermal effects can drive deep rock-slope deformation and progressive failure in the upper portion of the Randa rockslide (Gischig et al., 2011). Steeply plunging folds (Fig. Considering these information, it is thought that the new failure can develop in these slopes. Nevertheless, it is important to remember that geomechanical slope stability analyses are data-limited, meaning that modeling of the failure mechanism or slope design must be achieved with few site-specific data which are often not fully representative of the entire slope (Eberhardt et al., 2004; Itasca, 2004). 3. Two cases were considered for the analysis, Case (1) Tension crack is on the inclined free surface of rock slope and Case (2) tension crack is on the slope face. First, superimposed on the climate change effect is a background seismic signal that may produce CMFs unrelated to thermal changes in the slope environment (e.g., Tarr and Martin, 1912; van der Woerd etal., 2004; Post, 1967; Jibson etal., 2006; Allen etal., 2011). (2015) proposed a method combining a probabilistic approach (assuming the distribution of the rock bridges along the sliding plane follows a fractal distribution law) using the discrete element method (DEM), to investigate translational sliding failure along a single incipient discontinuity within rock slopes. (Servico Aerofotografco Nacional de Per photograph; January 9, 1962). This can be visualized by looking at Figure 6.10. The values input parameters, shear strength parameters, and depth of the water in the tension crack used in these analyses are obtained by probabilistic back-analyses. However, even in circumstances when knowledge about a historical event is well constrained, the interpretation of coarse rubbly deposits may not be straightforward (cf. Marc-Andr Brideau, Nicholas J. Roberts, in Landslide Hazards, Risks, and Disasters (Second Edition), 2022. I chose Mathcad for programming rather than some other software like Matlab, because I find it more visually instructive. New interpretation of the famous Bunzkgele case study (Austria) based on field work and simulation to update (Sausgruber et al., 2010) the interpretation by Zischinsky (1969). Horizontal folds may be associated with simple translational failure (sliding of rock blocks in the true dip direction) on one limb and toppling mechanisms on the other fold limb, depending on their orientation with respect to a given slope face (Fig. Hazard Mitigation: approaches to CMF hazard mitigation are well developed in some national jurisdictions (e.g., Switzerland) and with reference to selected CMF processes in sub-national regions (e.g., glacial lakes in the Cordillera Blanca, Peru; Portocarrero, 2014). These fractures can be mapped on aerial photographs for later, more detailed, measurement and analysis. The association of kinematic rock slope failure modes with anticlinal folding has been discussed by Badger (2002), who described the relationship between failure (planar/toppling) and the relative location of the rock slope and discontinuity sets commonly associated with folded, bedded units. The following two sections comprise one of those mathematical diversions I mentioned previously. 1A shows the failure of rock mass as plane failure.

5). In addition, the effect of probabilistic distribution of persistence was investigated using a parametric method. In this case and in 2-D, the fracture tensor becomes, The first integral is simply the histogram moment of order 2 of the histogram of fracture set radii. This suggests that linkages in the CMF-climate change paradigm are complex (cf. Both ways will get us to pretty much the same place, but the first (concentrating on fractures) is probably more important for fluid flow, whereas the second (concentrating on rock blocks between fractures) is probably more important for rock mass slope stability. The SLOPESIM code was utilized to find the paths of minimum SM and achieve probabilistic failure analysis of a jointed rock slope. Nevertheless, these simulations are not able to fully reproduce the structures developing within the slope. (2007) showed the application of hydro-mechanical distinct-element approach to analyze the efficacy of the deep drainage system constructed to stabilize the large Campo Vallemaggia landslide (Switzerland). Fig. Fig. Some attempt using distinct-element model start to show interesting results (Preh et al., 2003; Stead and Coggan, 2012; Stead et al., 2012). 3,544m asl) arrowed at K, located in the glacier foreland of the Chorabari Glacier. Landslides in bedrock include some of the most common (rockfalls) and most destructive (rock avalanches) slope processes. Cruden and Krahn (1978) described the presence of slickensides on folded bedding surfaces, as well as minor thrusts, which controlled the Frank Slide.

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